Q.E.D.

30 31 KnottIng Polygons a proof by paper folding It is very easy to construct equilateral triangles, squares, and regular hexagons in a large number of different ways. Regular pentagons are more tricky, but here is the simplest way to construct one Tie a knot into a piece of tape and pull on the ends until the knot is completely flat. Cut off the excess tape on both sides and you are left with a regular pentagon Why does this work Consider two regular pentagons sharing one side together with a piece of tape running through both of them below, left. If we fold the left pentagon onto the right along the common side, the paper strip will neatly align itself along one of the sides of the right pentagon. Therefore, if we keep folding the tape around this pentagon, we will successively define all its sides and diagonals. Unwrapping the now creased tape and discarding the pentagons, we can finally tie the tape into a knot and flatten it such that no new creases appear. The regular polygons with more than five sides can also be knotted using one or two paper strips, but the practical execution of these constructions can get very awkward.